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We obtain a finite analogue of a recent generalization of an identity inRamanujan's Notebooks. Differentiating it with respect to one of the parametersleads to a result whose limiting case gives a finite analogue of Andrews'famous identity for $\textup{spt}(n)$. The latter motivates us to extend thetheory of the restricted partition function $p(n, N)$, namely, the number ofpartitions of $n$ with largest parts less than or equal to $N$, by obtainingthe finite analogues of rank and crank for vector partitions as well as of therank and crank moments. As an application of the identity for our finiteanalogue of the spt-function, namely $\textup{spt}(n, N)$, we prove aninequality between the finite second rank and crank moments. The other resultsobtained include finite analogues of a recent identity of Garvan, an identityrelating $d(n, N)$ and lpt$(n, N)$, namely the finite analogues of the divisorand largest parts functions respectively, and a finite analogue of theBeck-Chern theorem. We also conjecture an inequality between the finiteanalogues of $k^{\textup{th}}$ rank and crank moments.

Untrodden pathways in the theory of the restricted partition function $p(n, N)$